At the end of last year before going away for the holidays, I laid out a miniature program for learning / teaching about rates of change. The first step was to appreciate the tremendous difference between linear and exponential change. It is easy to write down the relevant definitions: in linear growth we take the previous value and add a constant to it, whereas in exponential growth we take the previous value and multiply with a constant (that’s greater than 1 – for a constant less than 1 you get exponential decay).
It is also relatively easy to give real life examples. For instance, the distance traveled by a car driving on the highway at 60 miles per hour is an example of linear growth. It increases by 1 mile for every minute of driving. An example of exponential growth is compound interest. Suppose you are being paid 8% of interest, then each year you have 1.08 times your previous year’s capital.
But that’s where any similarity between the two types of growth ends. As humans, we simply lack the imagination for just how powerful exponential growth really is. This is captured by some great quotes and stories:
The earliest is the story from over 1,000 years ago of the inventor of the chess board, who when offered any prize by the King asked only for one grain of rice on the first field, 2 on the second, and doubling from there on. The King quickly agreed not realizing that this exponential growth results in 460 billion tons of rice on the last field (or about 1,000 times the global annual production).
A likely apocryphal quote by Einstein is that “compound interest is the most powerful force in the universe.”
Here is Professor Bartlett in a lecture: “[…] the greatest shortcoming of the human race is our inability to understand the exponential function.”
Why is that? Because exponential growth starts out looking not all that impressive. Here is is a plot of the exponential function y = e^x for values of x from 0 to 10 created using Wolfram Alpha
Given the scale of the y-axis, for value of x If that isn’t enough to convince you that this is outside of human intuition then please consider the next plot. This shows the *same* exponential function but this time for values of x from 0 to 20 Wait, what? Where did the explosive growth between 8 and 10 from the previous chart go? This curve looks entirely flat between 8 and 10 and in fact all the way to about 12! Well that is the nature of exponential growth. It continues to accelerate so much that as you zoom out, the further growth just makes the previous growth look tiny. And that completely defies all of our intuition. PS. Polynomial growth falls between linear and exponential. I may write a separate post about it in the future.
